| L(s) = 1 | + 0.321·3-s + 3.89·5-s − 7-s − 2.89·9-s − 11-s − 0.218·13-s + 1.25·15-s + 4.86·17-s + 0.643·19-s − 0.321·21-s + 6.53·23-s + 10.1·25-s − 1.89·27-s + 10.4·29-s + 3.03·31-s − 0.321·33-s − 3.89·35-s − 1.89·37-s − 0.0701·39-s − 4.86·41-s + 1.35·43-s − 11.2·45-s + 5.57·47-s + 49-s + 1.56·51-s − 7.79·53-s − 3.89·55-s + ⋯ |
| L(s) = 1 | + 0.185·3-s + 1.74·5-s − 0.377·7-s − 0.965·9-s − 0.301·11-s − 0.0605·13-s + 0.323·15-s + 1.17·17-s + 0.147·19-s − 0.0701·21-s + 1.36·23-s + 2.03·25-s − 0.364·27-s + 1.93·29-s + 0.545·31-s − 0.0559·33-s − 0.658·35-s − 0.311·37-s − 0.0112·39-s − 0.759·41-s + 0.206·43-s − 1.68·45-s + 0.813·47-s + 0.142·49-s + 0.218·51-s − 1.07·53-s − 0.525·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257116739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257116739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.321T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 13 | \( 1 + 0.218T + 13T^{2} \) |
| 17 | \( 1 - 4.86T + 17T^{2} \) |
| 19 | \( 1 - 0.643T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 + 8.01T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 8.86T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 9.72T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.746T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.818097819640863138716355750056, −8.952843581421267951464040812508, −8.331413749491139292666912151137, −7.09515376000597973995126072089, −6.24161159665288825981651303752, −5.58019951139022636755356693450, −4.87426731002190772717773418368, −3.12394140904715815531951414415, −2.60584075817615020448386719142, −1.20569383612057266266721391215,
1.20569383612057266266721391215, 2.60584075817615020448386719142, 3.12394140904715815531951414415, 4.87426731002190772717773418368, 5.58019951139022636755356693450, 6.24161159665288825981651303752, 7.09515376000597973995126072089, 8.331413749491139292666912151137, 8.952843581421267951464040812508, 9.818097819640863138716355750056
